Difference between revisions of "Inessential mapping"
From Encyclopedia of Mathematics
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''homotopically-trivial mapping'' | ''homotopically-trivial mapping'' | ||
| − | A continuous mapping | + | A continuous mapping $f : X \to Q^n$ of a topological space $X$ into the $n$-dimensional ball $Q^n$ such that there is a continuous mapping $g : X \to Q^n$ that coincides with $f$ on the inverse image $f^{-1} S^{n-1}$ of the boundary $S^{n-1}$ of $Q^n$ and takes $X$ into $S^{n-1}$ (that is, $gX \subseteq S^{n-1}$). When $X$ is a normal Hausdorff space, then $\dim X < n$ if and only if every continuous mapping $f : X \to Q^n$, $n = 1, 2, \dots ,$ is inessential (Aleksandrov's theorem). |
| − | A continuous mapping of a topological space into the | + | A continuous mapping of a topological space into the $n$-dimensional sphere is called inessential if it is homotopic to the constant mapping. |
====Comments==== | ====Comments==== | ||
| − | The term "homotopically-trivial mapping" is not used in the context | + | The term "homotopically-trivial mapping" is not used in the context $f : X \to Q^n$. See [[#References|[a1]]], [[#References|[a2]]] and [[#References|[a3]]]. |
Generally, a mapping which is homotopic to a constant mapping is called nullhomotopic or homotopically trivial; in [[#References|[a3]]] they are called inessential. See also [[Essential mapping|Essential mapping]]. | Generally, a mapping which is homotopic to a constant mapping is called nullhomotopic or homotopically trivial; in [[#References|[a3]]] they are called inessential. See also [[Essential mapping|Essential mapping]]. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.S. Aleksandrov, "Topologie" , '''1''' , Springer (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , PWN (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.S. Aleksandrov, "Topologie" , '''1''' , Springer (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Engelking, "Dimension theory" , PWN (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)</TD></TR></table> | ||
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Latest revision as of 06:26, 14 January 2017
homotopically-trivial mapping
A continuous mapping $f : X \to Q^n$ of a topological space $X$ into the $n$-dimensional ball $Q^n$ such that there is a continuous mapping $g : X \to Q^n$ that coincides with $f$ on the inverse image $f^{-1} S^{n-1}$ of the boundary $S^{n-1}$ of $Q^n$ and takes $X$ into $S^{n-1}$ (that is, $gX \subseteq S^{n-1}$). When $X$ is a normal Hausdorff space, then $\dim X < n$ if and only if every continuous mapping $f : X \to Q^n$, $n = 1, 2, \dots ,$ is inessential (Aleksandrov's theorem).
A continuous mapping of a topological space into the $n$-dimensional sphere is called inessential if it is homotopic to the constant mapping.
Comments
The term "homotopically-trivial mapping" is not used in the context $f : X \to Q^n$. See [a1], [a2] and [a3].
Generally, a mapping which is homotopic to a constant mapping is called nullhomotopic or homotopically trivial; in [a3] they are called inessential. See also Essential mapping.
References
| [a1] | P.S. Aleksandrov, "Topologie" , 1 , Springer (1974) |
| [a2] | R. Engelking, "Dimension theory" , PWN (1977) |
| [a3] | W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948) |
How to Cite This Entry:
Inessential mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inessential_mapping&oldid=40180
Inessential mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inessential_mapping&oldid=40180
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article